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[2011 question paper]
... Answer any five questions. All questions carry 10 marks 1. (a) Consider the ring R of polynomials in n variables with integer coefficients. Prove that the polynomial f (x1 , x2 , . . . , xn ) = x1 x2 · · · xn has 2n+1 − 2 non-constant polynomials in R dividing it. (b) Let p1 , p2 , . . . , pn be dis ...
... Answer any five questions. All questions carry 10 marks 1. (a) Consider the ring R of polynomials in n variables with integer coefficients. Prove that the polynomial f (x1 , x2 , . . . , xn ) = x1 x2 · · · xn has 2n+1 − 2 non-constant polynomials in R dividing it. (b) Let p1 , p2 , . . . , pn be dis ...
Algebra 2 - Identifying and Evaluating Functions
... Algebra 2 - Identifying and Evaluating Functions ____ ...
... Algebra 2 - Identifying and Evaluating Functions ____ ...
Lecture 4
... of B, written A B iff there is an injective (oneto-one) function f:A B. The size of a set A equals the size of the set B, written A B iff there is a bijective (one-to-one and onto) function f:A B. We say that the set A is equipotent (or equinumerous) with B. Note: If A is finite and ha ...
... of B, written A B iff there is an injective (oneto-one) function f:A B. The size of a set A equals the size of the set B, written A B iff there is a bijective (one-to-one and onto) function f:A B. We say that the set A is equipotent (or equinumerous) with B. Note: If A is finite and ha ...
Full text
... The brief tables above are compiled to show the effectiveness of the algorithm. For a fixed but arbitrary choice of n e P, we observe that: (1) to compute q(n) we need about -Jn of the values q(k), 0
... The brief tables above are compiled to show the effectiveness of the algorithm. For a fixed but arbitrary choice of n e P, we observe that: (1) to compute q(n) we need about -Jn of the values q(k), 0
Georg Cantor (1845
... usual arithmetic operations +,x,^ for them. He also believes that like numbers one can always compare two ordinals or two cardinals a and b in such a way that one of the following a
... usual arithmetic operations +,x,^ for them. He also believes that like numbers one can always compare two ordinals or two cardinals a and b in such a way that one of the following a
PDF
... Definition 2. For any number t, let varP (t) denote the number of sign changes in the sequence P0 (t), P1 (t), . . .. Theorem 1. For real numbers a and b that are both not roots of P (x), #{distinct real roots of P in (a, b)} = varP (a) − varP (b) In particular, we can count the total number of dist ...
... Definition 2. For any number t, let varP (t) denote the number of sign changes in the sequence P0 (t), P1 (t), . . .. Theorem 1. For real numbers a and b that are both not roots of P (x), #{distinct real roots of P in (a, b)} = varP (a) − varP (b) In particular, we can count the total number of dist ...
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... A1 Let f be a real-valued function on the plane such that for every square ABCD in the plane, f (A) + f (B) + f (C) + f (D) = 0. Does it follow that f (P ) = 0 for all points P in the plane? A4 Let S be a set of rational numbers such that (a) 0 ∈ S; (b) If x ∈ S then x + 1 ∈ S and x − 1 ∈ S; and (c) ...
... A1 Let f be a real-valued function on the plane such that for every square ABCD in the plane, f (A) + f (B) + f (C) + f (D) = 0. Does it follow that f (P ) = 0 for all points P in the plane? A4 Let S be a set of rational numbers such that (a) 0 ∈ S; (b) If x ∈ S then x + 1 ∈ S and x − 1 ∈ S; and (c) ...
PDF
... Theorem. If the real function f is continuous on the interval [0, ∞) and the limit lim f (x) exists as a finite number a, then f is uniformly continuous x→∞ on that interval. Proof. Let ε > 0. According to the limit condition, there is a positive number M such that ε ...
... Theorem. If the real function f is continuous on the interval [0, ∞) and the limit lim f (x) exists as a finite number a, then f is uniformly continuous x→∞ on that interval. Proof. Let ε > 0. According to the limit condition, there is a positive number M such that ε ...
Notes
... there are “holes” in the number line filled√by irrational numbers, such as 2. The Greeks discovered and proved that 2 is not rational. Exercise: find a very simple proof of this. They soon found that the square root of every prime is irrational. By the time of Euclid’s Elements (300 BCE) the irratio ...
... there are “holes” in the number line filled√by irrational numbers, such as 2. The Greeks discovered and proved that 2 is not rational. Exercise: find a very simple proof of this. They soon found that the square root of every prime is irrational. By the time of Euclid’s Elements (300 BCE) the irratio ...
Not enumerating all positive rational numbers
... and temporal axes it is clear already that bijections between different infinite sets are impossible too. But here we will give an independent proof. Cantor's enumeration of the set –+ positive rational numbers q is ordered by the ascending sum (a+b) of numerator a and denominator b of q = a/b, and ...
... and temporal axes it is clear already that bijections between different infinite sets are impossible too. But here we will give an independent proof. Cantor's enumeration of the set –+ positive rational numbers q is ordered by the ascending sum (a+b) of numerator a and denominator b of q = a/b, and ...
1.1 Real Numbers & Number Operations
... Examples of Real numbers • Whole numbers: 0, 1, 2, 3 (counting #s) • Integers: -2, -1, 0, 1, 2 (+ & - whole #s) • Rational numbers: a # that can be written as a fraction. When written as a decimal, they terminate or repeat. ½, 1/3, 4/5, 7/9 • Irrational numbers: real #s that are not rational such a ...
... Examples of Real numbers • Whole numbers: 0, 1, 2, 3 (counting #s) • Integers: -2, -1, 0, 1, 2 (+ & - whole #s) • Rational numbers: a # that can be written as a fraction. When written as a decimal, they terminate or repeat. ½, 1/3, 4/5, 7/9 • Irrational numbers: real #s that are not rational such a ...